Simplify and expand the following expression: $ \dfrac{1}{y + 3}+ \dfrac{5}{y + 10}+ \dfrac{5y}{y^2 + 13y + 30} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{5y}{y^2 + 13y + 30} = \dfrac{5y}{(y + 3)(y + 10)}$ Now we have: $ \dfrac{1}{y + 3}+ \dfrac{5}{y + 10}+ \dfrac{5y}{(y + 3)(y + 10)} $ The least common multiple of the denominators is: $ (y + 3)(y + 10)$ In order to get the first term over $(y + 3)(y + 10)$ , multiply by $\dfrac{y + 10}{y + 10}$ $ \dfrac{1}{y + 3} \times \dfrac{y + 10}{y + 10} = \dfrac{y + 10}{(y + 3)(y + 10)} $ In order to get the second term over $(y + 3)(y + 10)$ , multiply by $\dfrac{y + 3}{y + 3}$ $ \dfrac{5}{y + 10} \times \dfrac{y + 3}{y + 3} = \dfrac{5(y + 3)}{(y + 3)(y + 10)} $ Now we have: $ \dfrac{y + 10}{(y + 3)(y + 10)} + \dfrac{5(y + 3)}{(y + 3)(y + 10)} + \dfrac{5y}{(y + 3)(y + 10)} $ $ = \dfrac{ y + 10 + 5(y + 3) + 5y} {(y + 3)(y + 10)} $ Expand: $ = \dfrac{y + 10 + 5y + 15 + 5y}{y^2 + 13y + 30} $ $ = \dfrac{11y + 25}{y^2 + 13y + 30}$